Isolated singularities for semilinear elliptic systems with power-law nonlinearity

Abstract

We study the system $-\mathrm{\Delta }u=|u{|}^{\alpha -1}u$ with $1<\alpha \le \frac{n+2}{n-2}$, where $u=\left(u_1,\dots ,u_m\right)$, $m\ge 1$, is a $C^2$ nonnegative function that develops an isolated singularity in a domain of ${ℝ}^n$, $n\ge 3$. Due to the multiplicity of the components of $u$, we observe a new Pohozaev invariant different than the usual one in the scalar case. Aligned with the classical theory of the scalar equation, we classify the solutions on the whole space as well as the punctured space, and analyze the exact asymptotic behavior of local solutions around the isolated singularity. On a technical level, we adopt the method of moving spheres and the balanced-energy-type monotonicity functionals.